Okay now, I'm a little worried that my name appears in the above post and in the next paragraph you plan to use bleach.
Please, I do not use bleach. That's my personal choice due to an allergy.
But, I know from experience at work that ...pay attention...bleach does not kill mold or fungus . Nope. It does not kill it. There are other products you could use that will possibly kill what is growing in and on your unfinished wood, but all bleach will do is remove stains, temporarily slow the growth, and will ultimately weaken the wood itself.
Regular laundry bleach by itself and/or mixed with water will not even penetrate into the wood far enough to touch the stuff growing in there. To allow the bleach to travel you would need to add a surfactant - still and all you will not kill the stuff. Even industrial strength bleach can't kill mold/fungus on or in wood. You need a product that has antimicrobial and/or biocide ingredients; that will cost more than you paid for the table in the first place.
My advice is to remove and burn the wood on the top of the table; then replace the top with properly prepared wood if you intend to use the table as a place to store potted plants (wet environment). Or even better, replace the top with a framed metal grid/fencing to allow water to run off similar to what some greenhouses have as plant tables.